Journal of Fourier Analysis and Applications

, Volume 5, Issue 5, pp 465–494 | Cite as

Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion

  • Yves Meyer
  • Fabrice Sellan
  • Murad S. Taqqu


We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes Lévy's midpoint displacement technique which is used to generate Brownian motion. The low-frequency terms in the expansion involve an independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections. We also obtain a way of constructing an arbitrary number of non-Gaussian continuous time processes whose second order properties are the same as those of fractional Brownian motion.

Keywords and Phrases

Fractional ARIMA midpoint displacement technique fractional Gaussian noise fractional derivative generalized functions self-similarity 

Math Subject Classifications

Primary 60G18 secondary 41A58 60F15. 


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  1. [1]
    Abry, P., Flandrin, P. Taqqu, M.S., and Veitch, D. (1998). Wavelets for the analysis, estimation and synthesis of scaling data. In Park, K. and Willinger, W., Eds.,Self-Similar Network Traffic and Performance Evaluation, Wiley (Interscience Division), New York. To appear.Google Scholar
  2. [2]
    Abry, P. and Sellan, F. (1996). The wavelet-based synthesis for the fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation,Appl. Comp. Harmonic Anal.,3(4), 377–383.Google Scholar
  3. [3]
    Abry, P. and Veitch, D. (1997). Wavelet analysis of long range dependent traffic,IEEE Trans. Info. Theor., To appear.Google Scholar
  4. [4]
    Beran, J. (1994).Statistics for Long-Memory Processes, Chapman & Hall, New York.Google Scholar
  5. [5]
    Billingsley, P. (1968).Convergence of Probability Measures, John Wiley & Sons, New York.Google Scholar
  6. [6]
    Brockwell, P.J. and Davis, R.A. (1987).Time Series: Theory and Methods, Springer-Verlag, New York.Google Scholar
  7. [7]
    Chow, Y.S. and Teicher, H. (1988).Probability Theory: Independence, Interchangeability, Martingales, 2nd ed., Springer-Verlag, New York.Google Scholar
  8. [8]
    Cohen, A. and Ryan, R.D. (1995).Wavelets and Multiscale Signal Processing, Chapman & Hall, London.Google Scholar
  9. [9]
    Elliott. F.W. and Majda, A.J. (1994). A wavelet Monte Carlo method for turbulent diffusion with many spatial scales,J. Comp. Phys.,113(1), 82–111.Google Scholar
  10. [10]
    Flandrin, P. (1992). Wavelet analysis and synthesis of fractional Brownian motion,IEEE Trans. Info. Theor.,IT-38(2), 910–917.Google Scholar
  11. [11]
    Gelfand, I.M. and Shilov, G.E. (1964).Generalized Functions: Properties and Operations, vol. 1, Academic Press, New York.Google Scholar
  12. [12]
    Gelfand, I.M. and Vilenkin, N.Ya. (1964).Generalized Functions: Applications of Harmonic Analysis, vol.4, Academic Press, New York.Google Scholar
  13. [13]
    Heneghan, C., Lowen, S.B., and Teich, M.C. (1996). Two-dimensional fractional Brownian motion: wavelet analysis and synthesis,Proc. IEEE Southwest Symp. Image Anal. Interpretation, San Antonio.Google Scholar
  14. [14]
    Hernández, E. and Weiss, G. (1996).A First Course on Wavelets, CRC Press, Boca Raton, FL.Google Scholar
  15. [15]
    Houdré, C. (1993). Wavelets, probability and statistics: some bridges, In Benedetto, J.J. and Frazier, M.W., Eds.,Wavelets: Mathematics and Applications, CRC Press, Boca Raton, FL, 361–394.Google Scholar
  16. [16]
    Jaffard, S. and Meyer, Y. (1996).Wavelet Methods for Pointwise Regularity and Local Oscillations of Functions, vol. 587, American Mathematical Society, Providence, RI.Google Scholar
  17. [17]
    Kwapień, S. and Woyczyński, N.A. (1992).Random Series and Stochastic Integrals: Single and Multiple, Birkhäuser, Boston.Google Scholar
  18. [18]
    Lau, W.-C., Erramilli, A., Wang, J.L., and Willinger, W. (1995). Self-similar traffic generation: The random midpoint displacement algorithm and its properties, InProc. ICC '95, Seattle, WA, 466–472.Google Scholar
  19. [19]
    Leland, W.E., Taqqu, M.S., Willinger, W., and Wilson, D.V. (1994). On the self-similar nature of Ethernet traffic (Extended version),IEEE/ACM Trans. Networking,2, 1–15.Google Scholar
  20. [20]
    Lévy, P. (1948).Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, 1st ed., 2nd éd., 1965.Google Scholar
  21. [21]
    Mandelbrot, B.B. (1982).The Fractal Geometry of Nature. Freeman, W.W., and Co., Eds., New York.Google Scholar
  22. [22]
    Mandelbrot, B.B. and Van Ness, J.W. (1968). Fractional Brownian motions, fractional noises and applications,SIAM Rev.,10, 422–437.Google Scholar
  23. [23]
    Masry, E. (1993). The wavelet transform of stochastic processes with stationary increments and its application to fractional Brownian motion,IEEE Trans. Info. Theor.,39(1), 260–264.Google Scholar
  24. [24]
    Meyer, Y. (1992).Ondelettes, Hermann, Paris.Google Scholar
  25. [25]
    Meyer, Y. (1992).Wavelets and Operators, vol. 37. Cambridge University Press, Cambridge, UK.Google Scholar
  26. [26]
    Meyer, Y. (1993).Wavelets—Algorithms and Applications, SIAM Society for Industrial and Applied Mathematics, Philadelphia, PA.Google Scholar
  27. [27]
    Meyer, Y. and Coifman, R. (1997).Wavelets: Calderón-Zygmund and Multilinear Operators, vol. 48. Cambridge University Press, Cambridge, UK.Google Scholar
  28. [28]
    Norros, I. (1995). On the use of fractional Brownian motion in the theory of connectionless networks,IEEE J. Selected Areas Comm.,13, 953–962.Google Scholar
  29. [29]
    Pipiras, V. and Taqqu, M.S. (1998). Convergence of the Weierstrass-Mandelbrot process to fractional Brownian motion, preprint.Google Scholar
  30. [30]
    Ramanathan, J. and Zeitouni, O. (1991). On the wavelet transform of fractional Brownian motion,IEEE Trans. Info. Theor.,IT-37(4), 1156–1158.Google Scholar
  31. [31]
    Samorodnitsky, G. and Taqqu, M.S. (1994).Stable Non-Gaussian Processes: Stochastic Models with Infinite Variance, Chapman & Hall, New York.Google Scholar
  32. [32]
    Sellan, F. (1995). Synthèse de mouvements browniens fractionnaires à l'aide de la transformation par ondelettes,Comptes Rendus de l'Académie des Sciences de Paris,321, 351–358, Série I.Google Scholar
  33. [33]
    Sinai, Ya.G. (1976). Self-similar probability distributions,Theor. Prob. Appl.,21, 64–80.Google Scholar
  34. [34]
    Wornell, G. (1996).Signal Processing with Fractals: A Wavelet-Based Approach, Prentice-Hall, Upper Saddle River, NJ.Google Scholar
  35. [35]
    Zygmund, A. (1979).Trigonometric Series, Vols. I and II, Cambridge University Press, Cambridge.Google Scholar

Copyright information

© Birkhäuser 1999

Authors and Affiliations

  • Yves Meyer
    • 1
  • Fabrice Sellan
    • 2
    • 3
  • Murad S. Taqqu
    • 4
  1. 1.Département de MathématiquesEcole Normale Supérieure de CachanCachanFrance
  2. 2.Matra Systèmes et InformationFrance
  3. 3.Laboratoire Analyse et Modeles StochastiquesFrance
  4. 4.Department of Mathematics and StatisticsBoston UniversityBostonUSA

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